Chaotic time dependence in a disordered spin system

Stochastic Ising and voter models on Z(d) are natural examples of Markov pr ocesses with compact state spaces. When the initial state is chosen uniform ly at random, can it happen that the distribution at time t has multiple (s ubsequence) limits as t --> infinity? Yes for the d = 1 Voter Model with Ra ndom Rates (VMRR)- which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature - if the disorder distribution is heavy-tai led. No (at least in a weak sense) for the VMRR when the tail is light or d greater than or equal to 2, These results are based on an analysis of the "localization" properties of Random Walks with Random Rates. Mathematics Su bject Classification (1991): 60K35, 82B44.